Figure 1.
Dynamics of velocity $ (V_1, \; V_2) $ with 6 particles: {(a)} velocity $ V_1 $ for $ t\in [0, \; 3] $ ; {(b)} velocity $ V_2 $ for $ t\in [0, \; 3] $
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2019, 24(12): 6817-6835.
doi: 10.3934/dcdsb.2019168
Flocking of Cucker-Smale model with intrinsic dynamics
| 1. | Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China |
| 2. | Department of Mathematics, Suzhou University of Science and Technology, Suzhou 215009, China |
In this paper, we consider the flocking problem of modified continu- ous-time and discrete-time Cucker-Smale models where every agent has its own intrinsic dynamics with Lipschitz property. The dynamics of the models are governed by the interplay between agents' own intrinsic dynamics and Cucker-Smale coupling dynamics. Based on the explicit construction of Lyapunov functionals, we show that conditional flocking would occur. And then we study the relationship between the Lipschitz constant $ L $ of the intrinsic dynamics and exponent $ \beta $ measuring the strength of the interaction between agents when flocking occurs. We also give two examples to show flocking might not occur for enough large $ L $ or unconditionally for $ \beta>0 $. At last, we provide several numerical simulations to illustrate our theoretical results.
Mathematics Subject Classification:
Primary: 92D25, 92D50; Secondary: 91D30.
Citation:
Lining Ru, Xiaoping Xue. Flocking of Cucker-Smale model with intrinsic dynamics. Discrete and Continuous Dynamical Systems - B,
2019, 24
(12)
: 6817-6835.
doi: 10.3934/dcdsb.2019168
![]()
References:
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S. Ahn, H. Choi, S.-Y. Ha and H. Lee,
On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.
doi: 10.4310/CMS.2012.v10.n2.a10. |
| [2] |
S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51(2010), 103301, 17pp.
doi: 10.1063/1.3496895. |
| [3] |
H. Bae, Y.-P. Choi, S.-Y. Ha and M. Kang,
Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, Nonlinearity, 25 (2012), 1155-1177.
doi: 10.1088/0951-7715/25/4/1155. |
| [4] |
S. Bolouki and R. Malham$\acute{e}$, Theorems about ergodicity and class-ergodicity of chains with applications in known consensus models, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton), 2012.
doi: 10.1109/Allerton.2012.6483385. |
| [5] |
Y. Cao, W. Ren, D. Casbeer and C. Schumacher,
Finite-time connectivity-preserving consensus of networked nonlinear agents with unknown Lipschitz terms, IEEE Trans. Autom. Control, 61 (2016), 1700-1705.
doi: 10.1109/TAC.2015.2479926. |
| [6] |
J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani,
Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.
doi: 10.1137/090757290. |
| [7] |
I. D. Couzin, J. Krause, N. R. Franks and S. Levin,
Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), 513-516.
doi: 10.1038/nature03236. |
| [8] |
F. Cucker and J.-G. Dong,
Avoiding collisions in flocks, IEEE Trans. Autom. Control, 55 (2010), 1238-1243.
doi: 10.1109/TAC.2010.2042355. |
| [9] |
F. Cucker and J.-G. Dong,
On the critical exponent for flocks under hierarchical leadership., Math. Models Methods Appl. Sci., 19 (2009), 1391-1404.
doi: 10.1142/S0218202509003851. |
| [10] |
F. Cucker and E. Mordecki,
Flocking in noisy environments, J. Math. Pure Appl., 89 (2008), 278-296.
doi: 10.1016/j.matpur.2007.12.002. |
| [11] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Autom. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
| [12] |
F. Cucker and S. Smale,
On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.
doi: 10.1007/s11537-007-0647-x. |
| [13] |
F. Dalmao and E. Mordecki,
Cucker-Smale flocking under hierarchical leadership and random interactions, SIAM J. Appl. Math., 71 (2011), 1307-1316.
doi: 10.1137/100785910. |
| [14] |
F. Dalmao and E. Mordecki,
Hierarchical Cucker-Smale model subject to random failure, IEEE Trans. Autom. Control, 57 (2012), 1789-1793.
doi: 10.1109/TAC.2012.2188440. |
| [15] |
J.-G. Dong,
Flocking under hierarchical leadership with a free-will leader, Internat. J. Robust Nonlinear Control, 23 (2013), 1891-1898.
|
| [16] |
J.-G. Dong and L. Qiu,
Flocking of the Cucker-Smale Model on general digraphs, IEEE Trans. Autom. Control, 62 (2017), 5234-5239.
doi: 10.1109/TAC.2016.2631608. |
| [17] |
S.-Y. Ha, K. Lee and D. Levy,
Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.
doi: 10.4310/CMS.2009.v7.n2.a9. |
| [18] |
S.-Y. Ha and J.-G. Liu,
A simple proof of the Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.
doi: 10.4310/CMS.2009.v7.n2.a2. |
| [19] |
S.-Y. Ha and E. Tadmor,
From particle to kinetic and hydrodynamic description of flocking, Kinet. Relat. Models, 1 (2008), 415-435.
doi: 10.3934/krm.2008.1.415. |
| [20] |
A. Jadbabaie, J. Lin and A. Morse,
Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control, 48 (2003), 988-1001.
doi: 10.1109/TAC.2003.812781. |
| [21] |
C.-H. Li and S.-Y. Yang,
A new discrete cucker-smale flocking model under hierarchical leadership, Disc. Cont. Dyn. Syst. B, 21 (2016), 2587-2599.
doi: 10.3934/dcdsb.2016062. |
| [22] |
Z. Li,
Effectual leadership in flocks with hierarchy and individual preference, Disc. Cont. Dyn. Syst., 34 (2014), 3683-3702.
doi: 10.3934/dcds.2014.34.3683. |
| [23] |
Z. Li and S.-Y. Ha,
On the Cucker-Smale flocking with alternating leaders, Quart. Appl. Math., 73 (2015), 693-709.
doi: 10.1090/qam/1401. |
| [24] |
Z. Li, S.-Y. Ha and X. Xue,
Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership, Math. Models Methods Appl. Sci., 24 (2014), 1389-1419.
doi: 10.1142/S0218202514500043. |
| [25] |
Z. Li and X. Xue,
Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174.
doi: 10.1137/100791774. |
| [26] |
S. Motsch and E. Tadmor,
A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.
doi: 10.1007/s10955-011-0285-9. |
| [27] |
S. Motsch and E. Tadmor,
Heterophilious dynamics enhances consensus, SIAM Rev., 56 (2014), 577-621.
doi: 10.1137/120901866. |
| [28] |
R. Olfati-Saber, Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans. Autom. Control, 51 (2006), 401–420.
doi: 10.1109/TAC.2005.864190. |
| [29] |
L. Perea, P. Elosegui and G. Gómez,
Extension of the Cucker-Smale control law to space flight formation, Journal of Guidance, Control, and Dynamics, 32 (2009), 527-537.
doi: 10.2514/1.36269. |
| [30] |
C. Pignotti and I. Reche Vallejo,
Flocking estimates for the Cucker-Smale model with time lag and hierarchical leadership, Journal of Mathematical Analysis and Applications, 464 (2018), 1313-1332.
doi: 10.1016/j.jmaa.2018.04.070. |
| [31] |
L. Ru, Z. Li and X. Xue,
Cucker-Smale flocking with randomly failed interactions, Journal of the Franklin Institute–Engineering and Applied Mathematics, 352 (2015), 1099-1118.
doi: 10.1016/j.jfranklin.2014.12.007. |
| [32] |
J. Shen,
Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007/08), 694-719.
doi: 10.1137/060673254. |
| [33] |
C. Somarakis, E. Paraskevas, S. Baras and N. Motee, Synchronization and collision avoidance in non-linear flocking networks of autonomous agents, 24th Mediterranean Conference on Control and Automation (MED), June 21-24, 2016, Athens, Greece. |
| [34] |
C. Somarakis, E. Paraskevas, S. Baras and N. Motee,
Convergence analysis of classes of asymmetric networks of cucker-smale type with deterministic perturbations, IEEE Transactions on Control of Network Systems, 5 (2018), 1852-1863.
doi: 10.1109/TCNS.2017.2765824. |
| [35] |
H. Su, G. Chen, X. Wang and Z. Lin,
Adaptive second-order consensus of networked mobile agents with nonlinear dynamics, Automatica, 47 (2011), 368-375.
doi: 10.1016/j.automatica.2010.10.050. |
| [36] |
Y. Sun and W. Lin, A positive role of multiplicative noise on the emergence of flocking in a stochastic Cucker-Smale system, Chaos, 25 (2015), 083118, 7pp.
doi: 10.1063/1.4929496. |
| [37] |
Y. Sun, Y. Wang and D. Zhao,
Flocking of multi-agent systems with multiplicative and independent measurement noises, Physica A, 440 (2015), 81-89.
doi: 10.1016/j.physa.2015.08.005. |
| [38] |
T. V. Ton, N. T. H. Linh and A. Yagi,
Flocking and non-flocking behavior in a stochastic Cucker-Smale system, Anal. Appl., 12 (2014), 63-73.
doi: 10.1142/S0219530513500255. |
| [39] |
C. M. Topaz, A. L. Bertozzi and M. A. Lewis,
A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.
doi: 10.1007/s11538-006-9088-6. |
| [40] |
T. Vicsek, A. Czir$\acute{o}$k, E. Ben-Jacob, I. Cohen and O. Shochet,
Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.
doi: 10.1103/PhysRevLett.75.1226. |
| [41] |
T. Vicsek and A. Zefeiris,
Collective motion, Physics Reports, 517 (2012), 71-140.
doi: 10.1016/j.physrep.2012.03.004. |
| [42] |
W. Yu, G. Chen and M. Cao,
Consensus in directed networks of agents with nonlinear dynamics, IEEE Trans. Autom. Control, 56 (2014), 1436-1441.
doi: 10.1109/TAC.2011.2112477. |
| [43] |
W. Yu, G. Chen, M. Cao and J. Kurths,
Second-order consensus for multi-agent systems with directed topologies and nonlinear dynamics, IEEE Trans. Syst., Man, Cybern. B, Cybern., 40 (2010), 881-891.
|
show all references
References:
| [1] |
S. Ahn, H. Choi, S.-Y. Ha and H. Lee,
On collision-avoiding initial configurations to Cucker-Smale type flocking models, Commun. Math. Sci., 10 (2012), 625-643.
doi: 10.4310/CMS.2012.v10.n2.a10. |
| [2] |
S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51(2010), 103301, 17pp.
doi: 10.1063/1.3496895. |
| [3] |
H. Bae, Y.-P. Choi, S.-Y. Ha and M. Kang,
Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, Nonlinearity, 25 (2012), 1155-1177.
doi: 10.1088/0951-7715/25/4/1155. |
| [4] |
S. Bolouki and R. Malham$\acute{e}$, Theorems about ergodicity and class-ergodicity of chains with applications in known consensus models, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton), 2012.
doi: 10.1109/Allerton.2012.6483385. |
| [5] |
Y. Cao, W. Ren, D. Casbeer and C. Schumacher,
Finite-time connectivity-preserving consensus of networked nonlinear agents with unknown Lipschitz terms, IEEE Trans. Autom. Control, 61 (2016), 1700-1705.
doi: 10.1109/TAC.2015.2479926. |
| [6] |
J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani,
Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010), 218-236.
doi: 10.1137/090757290. |
| [7] |
I. D. Couzin, J. Krause, N. R. Franks and S. Levin,
Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), 513-516.
doi: 10.1038/nature03236. |
| [8] |
F. Cucker and J.-G. Dong,
Avoiding collisions in flocks, IEEE Trans. Autom. Control, 55 (2010), 1238-1243.
doi: 10.1109/TAC.2010.2042355. |
| [9] |
F. Cucker and J.-G. Dong,
On the critical exponent for flocks under hierarchical leadership., Math. Models Methods Appl. Sci., 19 (2009), 1391-1404.
doi: 10.1142/S0218202509003851. |
| [10] |
F. Cucker and E. Mordecki,
Flocking in noisy environments, J. Math. Pure Appl., 89 (2008), 278-296.
doi: 10.1016/j.matpur.2007.12.002. |
| [11] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Autom. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
| [12] |
F. Cucker and S. Smale,
On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.
doi: 10.1007/s11537-007-0647-x. |
| [13] |
F. Dalmao and E. Mordecki,
Cucker-Smale flocking under hierarchical leadership and random interactions, SIAM J. Appl. Math., 71 (2011), 1307-1316.
doi: 10.1137/100785910. |
| [14] |
F. Dalmao and E. Mordecki,
Hierarchical Cucker-Smale model subject to random failure, IEEE Trans. Autom. Control, 57 (2012), 1789-1793.
doi: 10.1109/TAC.2012.2188440. |
| [15] |
J.-G. Dong,
Flocking under hierarchical leadership with a free-will leader, Internat. J. Robust Nonlinear Control, 23 (2013), 1891-1898.
|
| [16] |
J.-G. Dong and L. Qiu,
Flocking of the Cucker-Smale Model on general digraphs, IEEE Trans. Autom. Control, 62 (2017), 5234-5239.
doi: 10.1109/TAC.2016.2631608. |
| [17] |
S.-Y. Ha, K. Lee and D. Levy,
Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.
doi: 10.4310/CMS.2009.v7.n2.a9. |
| [18] |
S.-Y. Ha and J.-G. Liu,
A simple proof of the Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325.
doi: 10.4310/CMS.2009.v7.n2.a2. |
| [19] |
S.-Y. Ha and E. Tadmor,
From particle to kinetic and hydrodynamic description of flocking, Kinet. Relat. Models, 1 (2008), 415-435.
doi: 10.3934/krm.2008.1.415. |
| [20] |
A. Jadbabaie, J. Lin and A. Morse,
Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control, 48 (2003), 988-1001.
doi: 10.1109/TAC.2003.812781. |
| [21] |
C.-H. Li and S.-Y. Yang,
A new discrete cucker-smale flocking model under hierarchical leadership, Disc. Cont. Dyn. Syst. B, 21 (2016), 2587-2599.
doi: 10.3934/dcdsb.2016062. |
| [22] |
Z. Li,
Effectual leadership in flocks with hierarchy and individual preference, Disc. Cont. Dyn. Syst., 34 (2014), 3683-3702.
doi: 10.3934/dcds.2014.34.3683. |
| [23] |
Z. Li and S.-Y. Ha,
On the Cucker-Smale flocking with alternating leaders, Quart. Appl. Math., 73 (2015), 693-709.
doi: 10.1090/qam/1401. |
| [24] |
Z. Li, S.-Y. Ha and X. Xue,
Emergent phenomena in an ensemble of Cucker-Smale particles under joint rooted leadership, Math. Models Methods Appl. Sci., 24 (2014), 1389-1419.
doi: 10.1142/S0218202514500043. |
| [25] |
Z. Li and X. Xue,
Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70 (2010), 3156-3174.
doi: 10.1137/100791774. |
| [26] |
S. Motsch and E. Tadmor,
A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.
doi: 10.1007/s10955-011-0285-9. |
| [27] |
S. Motsch and E. Tadmor,
Heterophilious dynamics enhances consensus, SIAM Rev., 56 (2014), 577-621.
doi: 10.1137/120901866. |
| [28] |
R. Olfati-Saber, Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans. Autom. Control, 51 (2006), 401–420.
doi: 10.1109/TAC.2005.864190. |
| [29] |
L. Perea, P. Elosegui and G. Gómez,
Extension of the Cucker-Smale control law to space flight formation, Journal of Guidance, Control, and Dynamics, 32 (2009), 527-537.
doi: 10.2514/1.36269. |
| [30] |
C. Pignotti and I. Reche Vallejo,
Flocking estimates for the Cucker-Smale model with time lag and hierarchical leadership, Journal of Mathematical Analysis and Applications, 464 (2018), 1313-1332.
doi: 10.1016/j.jmaa.2018.04.070. |
| [31] |
L. Ru, Z. Li and X. Xue,
Cucker-Smale flocking with randomly failed interactions, Journal of the Franklin Institute–Engineering and Applied Mathematics, 352 (2015), 1099-1118.
doi: 10.1016/j.jfranklin.2014.12.007. |
| [32] |
J. Shen,
Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007/08), 694-719.
doi: 10.1137/060673254. |
| [33] |
C. Somarakis, E. Paraskevas, S. Baras and N. Motee, Synchronization and collision avoidance in non-linear flocking networks of autonomous agents, 24th Mediterranean Conference on Control and Automation (MED), June 21-24, 2016, Athens, Greece. |
| [34] |
C. Somarakis, E. Paraskevas, S. Baras and N. Motee,
Convergence analysis of classes of asymmetric networks of cucker-smale type with deterministic perturbations, IEEE Transactions on Control of Network Systems, 5 (2018), 1852-1863.
doi: 10.1109/TCNS.2017.2765824. |
| [35] |
H. Su, G. Chen, X. Wang and Z. Lin,
Adaptive second-order consensus of networked mobile agents with nonlinear dynamics, Automatica, 47 (2011), 368-375.
doi: 10.1016/j.automatica.2010.10.050. |
| [36] |
Y. Sun and W. Lin, A positive role of multiplicative noise on the emergence of flocking in a stochastic Cucker-Smale system, Chaos, 25 (2015), 083118, 7pp.
doi: 10.1063/1.4929496. |
| [37] |
Y. Sun, Y. Wang and D. Zhao,
Flocking of multi-agent systems with multiplicative and independent measurement noises, Physica A, 440 (2015), 81-89.
doi: 10.1016/j.physa.2015.08.005. |
| [38] |
T. V. Ton, N. T. H. Linh and A. Yagi,
Flocking and non-flocking behavior in a stochastic Cucker-Smale system, Anal. Appl., 12 (2014), 63-73.
doi: 10.1142/S0219530513500255. |
| [39] |
C. M. Topaz, A. L. Bertozzi and M. A. Lewis,
A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.
doi: 10.1007/s11538-006-9088-6. |
| [40] |
T. Vicsek, A. Czir$\acute{o}$k, E. Ben-Jacob, I. Cohen and O. Shochet,
Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.
doi: 10.1103/PhysRevLett.75.1226. |
| [41] |
T. Vicsek and A. Zefeiris,
Collective motion, Physics Reports, 517 (2012), 71-140.
doi: 10.1016/j.physrep.2012.03.004. |
| [42] |
W. Yu, G. Chen and M. Cao,
Consensus in directed networks of agents with nonlinear dynamics, IEEE Trans. Autom. Control, 56 (2014), 1436-1441.
doi: 10.1109/TAC.2011.2112477. |
| [43] |
W. Yu, G. Chen, M. Cao and J. Kurths,
Second-order consensus for multi-agent systems with directed topologies and nonlinear dynamics, IEEE Trans. Syst., Man, Cybern. B, Cybern., 40 (2010), 881-891.
|
Figure 2.
Evolutions of $ d_V(t) $ and $ d_X(t) $ with 6 particles: {(a)} $ d_V(t) $ for $ t\in [0, \; 3] $ ; {(b)} $ d_X(t) $ for $ t\in [0, \; 3] $
Figure 3.
Dynamics of velocity $ (V_1, \; V_2) $ with 6 particles: {(a)} velocity $ V_1 $ for $ t\in [0, \; 150] $ ; {(b)} velocity $ V_2 $ for $ t\in [0, \; 150] $
Figure 4.
Evolutions of $ d_V[t] $ and $ d_X[t] $ with 6 particles: {(a)} $ d_V[t] $ for $ t\in [0, \; 150] $ ; {(b)} $ d_X[t] $ for $ t\in [0, \; 150] $
Figure 5.
Dynamics of velocity $ V_2 $ with 6 particles for: {(a)} $ \beta = 0.1 $ ; {(b)} $ \beta = 0.3 $ ; {(c)} $ \beta = 0.55 $ ; {(d)} $ \beta = 2.4 $
Figure 6.
Evolutions of $ d_V[t] $ with 6 particles for: {(a)} $ \beta = 0.1 $ ; {(b)} $ \beta = 0.3 $ ; {(c)} $ \beta = 0.55 $ ; {(d)} $ \beta = 2.4 $
Figure 7.
Evolutions of $ d_X[t] $ with 6 particles for: {(a)} $ \beta = 0.1 $ ; {(b)} $ \beta = 0.3 $ ; {(c)} $ \beta = 0.55 $ ; {(d)} $ \beta = 2.4 $
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